Formulation of proper and efficient algorithms for robot kinematics is essential for the analysis and design of serial manipulators. Kinematic modeling of manipulators is most often performed in Cartesian space. However, due to disadvantages of most widely used mathematical constructs for description of orientation such as Euler angles and rotational matrices, a need for unambiguous, compact, singularity free, computationally efficient method for representing rotational information is imposed. As a solution, unit quaternions are proposed and kinematic modeling in dual quaternion space arose. In this paper, an overview of spatial descriptions and transformations that can be applied together within these spaces in order to solve kinematic problems is presented. Special emphasis is on a different mathematical formalisms used to represent attitude of a rigid body such as rotation matrix, Euler angles, axis-angle representation, unit quaternions, and their mutual relation. Benefits of kinematic modeling in quaternion space are presented. New direct kinematics algorithm in dual quaternion space pertaining to a particular manipulator is given. These constructs and algorithms are demonstrated on the human centrifuge as 3 DoF robot manipulator.